200 ANALYTICAL MECHANICS PROBLEMS. 1. A reservoir which is 50 feet long, 40 feet wide, and 10 feet deep is full of water. Find the potential energy of the water relative to a plane 25 feet below the bottom of the reservoir. 2. A particle slides down a curve in a vertical plane and " loops the loop." Find the minimum height the starting point can have above the center of the "loop." The radius of the "loop" is 15 feet. 3. Find the least velocity with which a bullet will have to be projected from the earth so that it will never return again. 4. A uniform rod which is free to rotate about a horizontal axis is lield in a horizontal position. With what angular velocity will it pass the vertical position if it is let go? 6. A cylinder of mass m and radius a is rotating about a horizontal .axis, making n turns per second. How high can it raise a mass m', which is suspended from the cylinder by means of a string of negligible mass? 6. A particle, which is attached to a point by a string of negligible mass, has just enough energy to make complete revolutions in a vertical ^circle. Find the velocity at the highest and at the lowest points. 7. In the preceding problem show that the tension of the string is zero •when the particle is at the highest point and six times the weight of the particle when it is at the lowest point. 8. A particle starts from rest at the highest point of a smooth sphere and slides down under its own weight. Where will it leave the sphere? '9. A particle which is suspended by means of a string is pulled to one 'Side until it makes an angle a with the vertical, and then it is let go. Find the position at which the tension of the string equals the weight of the particle. 10. In the preceding problem show that the total energy remains constant during the motion of the particle. Also find the velocity at the lowest position when a = 60°. 11. Supposing the tensile force necessary to stretch an elastic string to be proportional to the increase in length, derive an expression for the potential energy of a stretched string. 12. Derive an expression for the potential energy of a watch spring.